Delays and structures pervade the realistic modeling of populations and their investigation under the paradigm of dynamical systems. They prove to be essential also in control and related fields, where modeling through delay functional or partial differential equations has become increasingly fundamental. The inclusion of past history in the time evolution and the introduction of structuring variables add nontrivial complexities with respect to ordinary systems, balancing the undoubted advantage of dealing with more realistic models. Equations involving time delays and structures both generate dynamical systems of infinite dimension, asking for advanced methods in the mathematical analysis and the numerical treatment. Finally, understanding stability of equilibria and other invariants is crucial and often requires sophisticated numerical and computational approaches.

The school brings together strong and up-to-date contributions in population dynamics and related fields as far as delays and structures give fundamental tools for the realistic modeling of, e.g., the transmission of an infectious disease, the evolution of a resource-consumer scenario or the competition in a predator-prey system. Numerical and computational expertise is also offered, providing reliable approaches towards a practical and accessible analysis. The course aims at discussing the most recent advances in the different contexts of the relevant mathematical analysis (functional aspects of semigroup theory); the concerned modeling approaches (delay differential, renewal and partial differential equations of evolution type, including multi-structured, neutral and state-dependent equations); the numerical and computational techniques to operate with infinite-dimensional dynamical systems (simulation, stability, bifurcation). This knowledge will be employed to discuss applications from ecology, epidemiology and life sciences in general. Laboratory sessions will allow the participants to learn both theoretical considerations and the practical application of modern software and packages (MATLAB/Octave, Python, MatCont, DDE-Biftool). Analysis, modeling, methods and applications will be illustrated focusing also on their interdisciplinary connections, starting from rapid introductions of the basics and reaching a state-of-the-art level by evolving classic approaches into modern perspectives.

The school is primarily addressed to PhD students and post-docs in the fields connected to structured population dynamics and dynamical systems involving time delays and their numerical analysis, ranging from mathematics to engineering and physics. Young and senior researchers in the above or neighboring fields, interested in gaining a compact yet comprehensive overview of population dynamics with delays and structures, are also welcome from academia or private R&D centers. The school also offers the possibility to learn and apply relevant software and computational tools through the investigation of case studies in the planned laboratory sessions.